Synchronous timing for interpolated timing recovery

ABSTRACT

Synchronous acquisition in a sampled amplitude read channel. A timing error estimation unit ( 302 ) is provided to calculate an acquisition timing error. The estimate is based on use of synchronous (interpolated) samples and an approximation to an arctangent function.

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application is related to application Ser. No. ______, titled “Asynchronous Timing for Interpolated Timing Recovery,” filed concurrently herewith.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to magnetic recording and, particularly, to an improved system and method for interpolated timing recovery.

[0004] 2. Description of the Related Art

[0005] In digital communications receivers, timing recovery circuits are used to acquire and then track the correct sampling and frequency of an analog signal. In an interpolated timing recovery circuit, this is accomplished by digitally resampling a stream of asynchronous samples of the analog signal. The digital resampling effectively reconstructs the values the signal takes on at points in time between the (asynchronous) times at which the signal was actually sampled. This is done by mathematically interpolating the asynchronous sample values of the signal. To allow for small errors in frequency, and to satisfy the Nyquist sampling criterion, the resampling period T_(s) is slightly longer than the asynchronous sampling period T_(a).

[0006] In the case of magnetic recording, as shown in FIG. 1, data sectors 100 on magnetic disks are formatted to include an acquisition preamble 102, a sync or synchronization mark 104, and user data 106. Timing recovery uses the acquisition preamble 102 to acquire the correct sampling frequency and phase before reading the user data 106. The synchronization mark 104 demarcates the beginning of the user data. The preamble pattern is periodic, having period 4T_(s), where T_(s) is the bit period.

[0007] The phase and frequency of the initial asynchronous samples of this waveform are unknown. The sampling phase (modulo the bit period T_(s)) can be anything, and the sampling frequency, can be in error by as much as half a percent. Interpolated timing recovery includes an asynchronous phase to estimate the initial sampling phase and to initialize the interpolator appropriately using the estimate. Then, a synchronous acquisition step is used to refine the initial estimate of the phase and correct the sampling frequency.

[0008] This is illustrated more clearly with reference to FIG. 2. In particular, FIG. 2 illustrates various initial sampling phases of the 2T acquisition preamble. Points y_(k) on the curve 1000 are the asynchronous samples, sampled at a sampling period of T_(a). Points z_(k) represent the interpolated (synchronous) points, at a resampling period of T_(s). The values μ_(k) represent the fractions of the asynchronous period T_(a) at which to interpolate the next synchronous sample. The objective of zero phase restart (or asynchronous sampling) is to determine an initial interpolation interval μ₀, i.e., the fraction of an asynchronous period T_(a) after the last asynchronous sample y_(k) at which to interpolate the first synchronous sample.

SUMMARY OF THE INVENTION

[0009] One aspect of the present invention relates to an improved asynchronous sampling system and method, i.e., an improved zero phase restart system. Another aspect of the invention relates to synchronous acquisition, i.e., determining the phase error estimate. A timing error estimation unit is provided to calculate an acquisition timing error. The estimate is based on use of synchronous (interpolated) samples and an approximation to an arctangent function.

[0010] An interpolated timing recovery system according to the present invention is simpler to implement, requires fewer signal samples, and is more robust against signal distortions such as gain errors, DC offset errors, and magneto-resistive asymmetry.

BRIEF DESCRIPTION OF THE DRAWINGS

[0011] A better understanding of the invention is obtained when the following detailed description is considered in conjunction with the following drawings in which:

[0012]FIG. 1 is a diagram of an exemplary data format of user data;

[0013]FIG. 2 is a diagram illustrating an exemplary 2T preamble and various phases;

[0014]FIG. 3 is a block diagram of an exemplary read/write channel according to an embodiment of the invention;

[0015]FIG. 4 is a block diagram of an exemplary interpolated timing recovery unit according to an embodiment of the invention;

[0016]FIG. 5 is a diagram of an exemplary zero phase restart unit according to an embodiment of the invention;

[0017]FIG. 6 is a table of exemplary hysteresis values;

[0018]FIG. 7 is a diagram of an exemplary timing error calculation unit according to an embodiment of the invention;

[0019]FIG. 8 is a diagram of an exemplary ITR filter according to an embodiment of the invention; and

[0020]FIG. 9 is a table of coefficient values for the interpolation filter of FIG. 8.

DETAILED DESCRIPTION OF THE INVENTION

[0021] FIGS. 2-9 illustrate an improved interpolated timing recovery system and method according to the present invention. Briefly, the present invention relates to an improved method for estimating the initial interpolation interval. Rather than employing an extensive series of relatively complex and time-consuming calculations, a look-up table technique is employed. The present invention further relates to determining a phase error estimate during synchronous acquisition.

[0022] Sampled Amplitude Read Channel

[0023] A block diagram of a sampled amplitude read channel according to an embodiment of the invention is shown in FIG. 3 and identified by the reference numeral 200. During a write operation, data is written onto the media. The data is encoded in an encoder 202, such as an RLL or other encoder. A precoder 204 precodes the sequence to compensate for the transfer function of the magnetic recording channel 208 and equalizing filters. The write circuitry 206 modulates the current in the recording head coil to record a binary sequence onto the medium. A reference frequency f_(ref) provides a write clock to the write circuitry 206.

[0024] The bit sequence is then provided to a variable gain amplifier 210 to adjust the amplitude of the signal. DC offset control 212 and loop filter/gain error correction 214 may be provided to control the adjustment of the VGA 210. Further, an asymmetry control unit 215 including an asymmetry adjustment unit 216 and asymmetry control 218 may be provided to compensate for magneto-resistive asymmetry effects.

[0025] The signal is then provided to a continuous time filter 220, which may be a Butterworth filter, for example, to attenuate high frequency noise and minimize aliasing into baseband after sampling. The signal is then provided to an analog to digital converter 222 to sample the output of the continuous time filter 220.

[0026] A finite impulse response filter 224 provides additional equalization of the signal to the desired response. The output of the FIR 224 is provided to an interpolated timing recovery unit 228 according to the present invention which is used to recover the discrete time sequence. The output of the interpolated timing recovery unit is used to provide a feedback control to the DC offset control 212, the gain error 214, the asymmetry control 218 and the FIR 224 control 226. The output of the interpolated timing recovery 228 is provided to a Viterbi detector 232 to provide maximum likelihood detection. The ITR output is provided to a sync detector 234 which is used to detect the sync mark using phase information gleaned from having read the immediately preceding preamble. This information is then provided to the Viterbi detector 232 for use in sequence detection. The Viterbi detector output is then provided to the decoder 236 which decodes the encoding provided by the encoder 202.

[0027] Interpolated Timing Recovery Unit

[0028] Acquisition timing according to the present invention is accomplished in the interpolated timing recovery unit 228. An exemplary interpolated timing recovery unit according to the present invention is shown in FIG. 4. The ITR unit 228 receives interpolated data y(t), which is provided to an estimator/slicer 300 and to a timing error estimator 302. An exemplary estimator/slicer 300 is described in commonly-owned U.S. patent application Ser. No. ______,titled “An Acquisition Signal Error Estimator,” filed Jan. 10, 2000 which is hereby incorporated by reference in its entirety as if fully set forth herein.

[0029] As will be explained in greater detail below, during timing acquisition, the timing error estimator 302 determines an estimated phase error between the actual interpolated sample time and the ideal interpolated sample time. The timing error estimator 302 provides its timing error output to a loop filter 304. The loop filter 304 filters the phase error to generate a frequency offset that settles to a value proportional to a frequency difference between the synchronous and asynchronous frequencies. The loop filter 304 provides its output to the time accumulator and phase calculation unit 306. The time accumulator and phase calculation unit 306 determines succeeding values for the interpolation interval μ_(k). The output of the time accumulator and phase calculator 306 is provided to the ITR filter 310.

[0030] As will be explained in greater detail below, a zero phase restart unit 308 according to the present invention is provided for determining an initial interpolation interval μ₀. The zero phase restart circuit 308 provides its output μ₀ to the time accumulator and phase calculator 306.

[0031] Loop Filter

[0032] The loop filter 304 may take a variety of forms. One particular implementation of the loop filter 304 employs a proportional and integral term (PI) of the form: ${L(z)} = {A + \frac{B}{\left( {1 - z^{- 1}} \right)}}$

[0033] It is noted, however, that the loop filter may take a variety of other forms. Thus, the form shown above is exemplary only.

[0034] Time Accumulator and Phase Calculation

[0035] The time accumulator and phase calculation unit 306 accumulates the frequency offset signal at the output of the loop filter 304: $\mu_{i} = {\sum\limits_{k \leq i}{\Delta \quad \mu_{k}}}$

[0036] where Δμ_(k) is the output of the loop filter and μ_(i) is, the fractional part of the cumulative sum of the μ_(k).

[0037] Asynchronous Acquisition (Zero Phase Restart)

[0038] The zero phase restart unit 308 computes an initial resampling phase μ₀ from a sequence of four successive asynchronous samples y₀, y₁, y₂, and y₃. The following formula approximates the phase (relative to the period 4T_(s), sinusoidal preamble signal) of the last asynchronous sample y₃: $\phi = {{\tan^{- 1}\left( \frac{y_{2} - y_{0}}{y_{3} - y_{1}} \right)} - {\delta \quad \pi}}$

[0039] where δ=(T_(s)−T_(a))/T_(a) is the oversampling margin, and the synchronous samples are ideally taken at the phases 0, π/2, π, and 3π/2 as is the case for any equalization target where the 2T preamble pattern is sampled at zero crossings. Then, an approximation for μ₀, the fraction of an asynchronous period T_(a) after the last asynchronous sample μ₃ at which to interpolate the first synchronous sample, is made. The approximation μ₀ is given by:

μ₀=(1−2φ/π)(1+δ)  (mod 1)

[0040] where 0≦μ₀<1.

[0041] If the above equation for φ is used, then μ₀ depends on the asynchronous data stream only through the quotient (y₂−y₀)/(y₃−y₁). As such, μ₀ may be approximated with a look-up table that is addressed according to the outcome of several comparisons of the form

a(y ₂ −y ₀)<b(y ₃ −y ₁)

[0042] where a and b are small integers. In one embodiment, the quotient is compared to each of the fractions b/a: {fraction (1/16)}, ⅛, ¼, ⅜, ½, ¾, 1, {fraction (4/3)}, 2, {fraction (8/3)}, 4, 8, and 16: The fractions are chosen for computational simplicity and to minimize the maximum error in the calculated value of μ₀ over its range of values. Then one of fourteen (14) values of μ₀ is returned by the lookup operation, according to which of the fourteen intervals (having these thirteen boundary points) the quotient falls in. For example, the table can be designed so as to output the average of the two values of 1 calculated using the two endpoints b₁/a₁ and b₂/a₂ whenever the quotient (y₂−y₀)/(y₃−y₁) falls anywhere between these two endpoints. Using this procedure with δ=0.0625 and rounding μ to eight unsigned bits, the following table (normalized to 256=1) is obtained: (45, 34, 18, 254, 235, 210, 182, 158, 130, 105, 86, 66, 50, 39).

[0043] This method of approximating μ by first approximating the quotient has the virtue of being robust against both gain and offset errors in the received signal.

[0044] An exemplary implementation of the zero phase restart unit 308 is shown in FIG. 5. As illustrated, a signal stream y_(i) is provided to a pair of delay operators 502, 504 and then an arithmetic operator 506. The output of the arithmetic operator 506 is y_(i)−y_(i−2). Along one branch, the output of the arithmetic operator 506 is provided to a signum operator 510 and then a delay operator 512. The output of the delay operator 512 and the signum operator 510 are provided to a multiplier 514. As will be described in greater detail below, the output of the multiplier 514 is used to determine the sign of The output of the arithmetic operator 506 is provided to an absolute value unit 508. The output of the absolute value unit 508 is provided to a bank of multipliers 520, which multiply |y_(i)−y_(i−2)| with the factors b_(n). The output of the absolute value unit is also provided to another delay operator 509, the output of which, |y_(i−1)−y_(i−3)| is provided to a multiplier bank 518. The multiplier bank 518 multiplies the output |y_(i−1)−y_(i−3)| with the factors a_(n). The resulting outputs of the multiplier banks 518, 520, are provided to a comparator bank 522. The comparators of the comparator bank 522 perform the comparisons a |y_(i−1)−y_(i−3)|<b|y_(i)−y_(i−2)|. The results of the compare operations are provided to a bank of AND gates 524, which select the appropriate value for |μ₀| from the lookup table 526. The signed value for μ₀ to is determined by multiplying the output of the multiplier 514 with |μ₀| in the multiplier 516.

[0045] Synchronous Acquisition

[0046] During synchronous acquisition, the phase error estimate _(τl) is calculated using four successive synchronous samples z_(i), z_(i+1), z_(i+2), z_(i+3). These samples are interpolated values of the signal. For EPR4 equalization, each of these samples should sample the period 4T_(s) sinusoidal preamble at one of the phases 0, λ/2, π, and 3π/2. The phase error estimate is calculated as an approximation to the formula ${\Delta \quad \tau_{i}} = \left\{ \begin{matrix} {\tan^{- 1}\left( \frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} < {{z_{i + 3} - z_{i + 2}}}} \\ {\tan^{- 1}\left( \frac{z_{i + 3} - z_{i + 1}}{z_{i + 2} - z_{i}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} \geq {{z_{i + 3} - z_{i + 1}}}} \end{matrix} \right.$

[0047] In one embodiment, the arctangent function is approximated by the identity function, and the quotient is approximated by a table lookup. A fast implementation approximates the magnitude of the quotient (z_(i+2)−z_(i))/(z_(i+3)−z_(i+1)) as follows: ${\frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}}} = \left\{ \begin{matrix} 0 & {if} & {{{{z_{i + 2} - z_{i}}} < {2^{- n}{{z_{i + 3} - z_{i + 1}}}}}\quad} \\ 2^{- k} & {if} & {\quad {{2^{- k}{{z_{i + 3} - z_{i + 1}}}} \leq {{z_{i + 2} - z_{i}}} < {{2^{{- k} + 1}{{z_{i + 3} - z_{i + 1}}}}}}\quad} \end{matrix} \right.$

[0048] where n is a fixed positive integer and i<=k<=n. The sign of the phase error estimate can be computed as the exclusive OR p′_(i)⊕q′_(i)⊕r′_(i) of the three Boolean variables:

p _(i)=(0≦z _(i+3) −z _(i+1))

q _(i)=(0<z _(i+2) −z _(i))

r _(i)=(|z _(i+2) −z _(i) |<|z _(i+3) −z _(i+1)|)

[0049] If p′_(i)q′_(i)⊕r′_(i) is true, then the sign is negative.

[0050] If, for whatever reason, the interpolated sampling phase error is close to T_(s)/2 in magnitude, then the sign of the phase error estimate |Δτ_(l)| is determined largely by noise. This can cause the correction to the phase to be essentially random in sign, which in turn causes the phase to remain at an unstable equilibrium value for an unacceptable period of time. This effect is known as hangup. To prevent hangup, the timing phase error estimate Δτ_(l) is adjusted under certain circumstances. This adjustment occurs when |Δτ_(l)| exceeds a fixed threshold t and the ‘closest’ sampling phase (according to the differences z_(i+2)−z_(i) and z_(i+3)−z_(i+1)) differs from the phase predicted by the hysteresis state h_(i) (i.e., a memory of the past sampling phase—an unsigned two bit value: 0, 1, 2, or 3) by −π/2 or π/2. Under these circumstances, Δτ_(l) is adjusted according to the present hysteresis state and the values of the Boolean variables p_(i), q_(i), and r_(i), defined above. Specifically, Δτ_(l) is adjusted by adding to it the value δ₁ when ═Δτ_(l)|>t and the variables h_(i), p_(i), q_(i), and r_(i), take one of the combinations of values specified in the table of FIG. 6. Under these circumstances, the hysteresis state is updated according to the table. Otherwise, the next hysteresis state is computed as the unsigned two bit value

h _(i+1) =r _(i)+2((q _(i)

r′ _(i))

(

p

_(i)

r _(i))).

[0051] An exemplary timing error calculation unit 302 is shown in FIG. 7. As illustrated, a signal stream z_(i) is provided to a pair of delay operators and then an arithmetic operator 706. The output of the arithmetic operator 706 is z_(i)−z_(i−2). Along one branch, the output of the delay operator is provided to an absolute value operator 708. As will be explained in greater detail below, along a second branch, the output of the arithmetic operator 706 is provided to a delay operator 710 and an XOR gate 712.

[0052] Along the first branch, the output of the absolute value operator 708 is then provided to a delay operator 714 and also to a bank of comparators 718. The output of the delay operator 714 is provided to a bank of multipliers 716, which multiply |z_(i−1)−z_(i−3)| by the powers of 2. The outputs of the multipliers 716 are provided as the other inputs to the comparators 718. The results of the compare operations are provided to a lookup table 714 which determines the value of |Δτ|.

[0053] As noted above, the output of the arithmetic operator 706 is provided to delay operator 710 and to XOR gate 712. The XOR gate 712 also receives as an input the output of an initial comparator 718 a. The comparator 718 a performs a comparison of |z_(i−1)−z_(i−3)| and |z_(i)−z_(i−2)|. The output of XOR gate 712 is provided, along with the output of the lookup table 714 to a multiplier 716 to define the sign of Δτ.

[0054] Interpolation Filter

[0055] As discussed above, once the estimates for μ are generated, they are provided to an interpolation filter 310 which generates the signal estimates. It can be shown that the interpolated samples are related to the received stream according to the following equation: $z_{k} = {\sum\limits_{i = {{- l} + 1}}^{l}\quad {{c_{i}\left( \mu_{k} \right)}y_{k + i}}}$

[0056] where the 2I interpolation coefficients ci(Pi are looked up in a table according to the value of 1. An exemplary interpolation filter 310 implementing the above equation is shown in FIG. 8 and FIG. 9. In particular, FIG. 8 shows the filter, and FIG. 9 is a table of exemplary lookup table values. Turning now to FIG. 8, the interpolation filter 310 receives a bit stream into a bank of delay operators 802. The outputs of the delay operators 802 are provided to multipliers 804 which multiply with the table 806 outputs. The results are summed back in arithmetic operator 808. 

What is claimed:
 1. An interpolated timing recovery unit in a sampled amplitude read channel, comprising: a timing error calculation unit configured to estimate a timing error by estimating the function ${\Delta \quad \tau_{i}} = \left\{ {\begin{matrix} {\tan^{- 1}\left( \frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} < {{z_{i + 3} - z_{i + 2}}}} \\ {\tan^{- 1}\left( \frac{z_{i + 3} - z_{i + 1}}{z_{i + 2} - z_{i}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} \geq {{z_{i + 3} - z_{i + 1}}}} \end{matrix}.} \right.$


2. An interpolated timing recovery unit according to claim 1, said timing error calculation unit configured to estimate a quotient ${\frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}}} = \left\{ \begin{matrix} 0 & {if} & {{{{z_{i + 2} - z_{i}}} < {2^{- n}{{z_{i + 3} - z_{i + 1}}}}}\quad} \\ 2^{- k} & {if} & {\quad {{2^{- k}{{z_{i + 3} - z_{i + 1}}}} \leq {{z_{i + 2} - z_{i}}} < {{2^{{- k} + 1}{{z_{i + 3} - z_{i + 1}}}}}}\quad} \end{matrix} \right.$

where n is a fixed positive integer and i<=k<=n and where z_(i), z_(i+1), z_(i+2), z_(i+3) are four successive synchronous samples.
 3. An interpolated timing recovery unit according to claim 2, wherein said timing error calculation unit uses a lookup table to estimate said timing error.
 4. A timing error calculation unit configured to determine an estimate for a timing error by estimating the function ${\Delta \quad \tau_{i}} = \left\{ {\begin{matrix} {\tan^{- 1}\left( \frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} < {{z_{i + 3} - z_{i + 2}}}} \\ {\tan^{- 1}\left( \frac{z_{i + 3} - z_{i + 1}}{z_{i + 2} - z_{i}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} \geq {{z_{i + 3} - z_{i + 1}}}} \end{matrix}.} \right.$


5. A timing error calculation unit according to claim 4, wherein said timing error calculation unit estimate a quotient ${\frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}}} = \left\{ \begin{matrix} 0 & {if} & {{{{z_{i + 2} - z_{i}}} < {2^{- n}{{z_{i + 3} - z_{i + 1}}}}}\quad} \\ 2^{- k} & {if} & {\quad {{2^{- k}{{z_{i + 3} - z_{i + 1}}}} \leq {{z_{i + 2} - z_{i}}} < {{2^{{- k} + 1}{{z_{i + 3} - z_{i + 1}}}}}}\quad} \end{matrix} \right.$

where n is a fixed positive integer and i<=k<=n and where z_(i), z_(i+1), z_(i+2), z_(i+3) are four successive synchronous samples.
 6. A timing error calculation unit according to claim 5, wherein said timing error calculation unit uses a lookup table to estimate said timing error.
 7. A method for interpolated timing recovery unit in a sampled amplitude read channel, comprising: estimating a timing error according to ${\Delta \quad \tau_{i}} = \left\{ \begin{matrix} {\tan^{- 1}\left( \frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} < {{z_{i + 3} - z_{i + 2}}}} \\ {\tan^{- 1}\left( \frac{z_{i + 3} - z_{i + 1}}{z_{i + 2} - z_{i}} \right)} & {if} & {{{z_{i + 2} - z_{i}}} \geq {{z_{i + 3} - z_{i + 1}}}} \end{matrix} \right.$

 where the arctangent function is approximated by the identity function.
 8. A method according to claim 7, comprising estimating a quotient: ${\frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}}} = \left\{ \begin{matrix} {{0\quad {if}\quad {{z_{i + 2} - z_{i}}}} < {2^{- n}{{z_{i + 3} - z_{i + 1}}}}} \\ {{2^{- k}\quad {if}\quad 2^{- k}\quad {{z_{i + 3} - z_{i + 1}}}} \leq {{z_{i + 2} - z_{i}}} < {{2^{{- k} + 1}{{z_{i + 3} - z_{i + 1}}}}}} \end{matrix} \right.$

where n is a fixed positive integer and i<=k<=n and where z_(i), z_(i+1), z_(i+2), z_(i+3) are four successive synchronous samples.
 9. A method according to claim 8, comprising using a lookup table to estimate said timing error.
 10. An interpolated timing recovery unit in a sampled amplitude read channel, comprising: a timing error calculation unit configured to estimate a timing error according to an estimate of an arctangent function.
 11. An interpolated timing recovery unit according to claim 10, said timing error calculation unit configured to estimate a timing error approximately according to ${\Delta \quad \tau_{i}} = \left\{ \begin{matrix} {{{{\tan^{- 1}\left( \frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}} \right)}\quad {if}\quad {{z_{i + 2} - z_{i}}}} < {{z_{i + 3} - z_{i + 2}}}}\quad} \\ {{{\tan^{- 1}\left( \frac{z_{i + 3} - z_{i + 1}}{\quad {z_{i + 2} - z_{i}}} \right)}\quad {if}\quad {{z_{i + 2} - z_{i}}}} \geq {{z_{i + 3} - z_{i + 1}}}} \end{matrix} \right.$

 where the arctangent function is approximated by the identity function.
 12. An interpolated timing recovery unit according to claim 11, said timing error calculation unit configured to estimate a quotient ${\frac{z_{i + 2} - z_{i}}{z_{i + 3} - z_{i + 1}}} = \left\{ \begin{matrix} {{0\quad {if}\quad {{z_{i + 2} - z_{i}}}} < {2^{- n}{{z_{i + 3} - z_{i + 1}}}}} \\ {{2^{- k}\quad {if}\quad 2^{- k}\quad {{z_{i + 3} - z_{i + 1}}}} \leq {{z_{i + 2} - z_{i}}} < {{2^{{- k} + 1}{{z_{i + 3} - z_{i + 1}}}}}} \end{matrix} \right.$

where n is a fixed positive integer and i<=k<=n and where z_(i), z_(i+1), z_(i+2), z_(i+3) are four successive synchronous samples. 